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Ampleness of Schur powers of cotangent bundles and p-hyperbolicity

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Abstract

In this paper, we study a variation of a conjecture of Debarre on positivity of cotangent bundles of complete intersections. We establish the ampleness of Schur powers of cotangent bundles of generic complete intersections in projective manifolds, with high enough explicit codimension and multi-degrees. Our approach is naturally formulated in terms of flag bundles and allows one to reach the optimal codimension. On complex manifolds, this ampleness property implies intermediate hyperbolic properties. We give a natural application of our main result in this context.
arXiv:1907.09174v1 [math.AG] 22 Jul 2019
AMPLENESS OF SCHUR POWERS OF COTANGENT BUNDLES
AND p-HYPERBOLICITY
ANTOINE ETESSE
Introduction
In [Deb05], Debarre proved that complete intersections of sufficiently ample general
hypersurfaces in complex Abelian varieties whose codimensions are at least as large as
their dimensions have ample cotangent bundles. By analogy, he suggested that this result
should also hold when abelian varieties are replaced by complex projective spaces. This
has been recently proved by Xie [Xie18] and Brotbek–Darondeau [BD18], independently,
based on ideas and explicit methods developped in [Bro16]. It is well-known that complex
compact manifolds with ample cotangent bundles are complex hyperbolic in the sense of
Kobayashi (Cf. [Kob13]).
More generally, for p1, the notion of p-infinitesimal hyperbolicity (See Section 5)
generalizes the usual notion of hyperbolicity. Indeed, Kobayashi hyperbolicity is equiv-
alent to 1-infinitesimal hyperbolicity. For complex compact manifolds, 1-infinitesimal
hyperbolicity implies p-infinitesimal hyperbolicity. Thus, the ampleness of the cotangent
bundle implies p-infinitesimal hyperbolicity. However, to obtain this weaker property, it
is sufficient to assume the ampleness of the pth exterior power of the cotangent bundle
([Dem, Prop. 3.4]). In this work, we will extend and generalize the results of [BD18]
in order to obtain ampleness of Schur powers of cotangent bundles of generic complete
intersections inside projective spaces.
Exterior powers are examples of Schur powers. These are obtained by applying to
vector bundles certain Schur functors associated to partitions λ= (λ1 · · · λk>0)
(see Sect. 1 for more details). We can actually prove the following ampleness result for
such general Schur powers.
Main Theorem. Let Mbe a projective variety of dimension Nover an algebraically
closed field kof characteristic zero, equipped with a fixed very ample line bundle OM(1)
M. Let cNbe a positive integer.
For any partition λ= (λ1 · · · λk>0) with kparts such that (k+ 1)cN, the
λth Schur power of the cotangent bundle of a complete intersection X:=H1∩ · · · Hc
of cgeneric hypersurfaces Hi∈ |OM(di)|with respective degrees disuch that
di1 + 2c2λ1+λ2+···+λk
gcd(λ1,...,λk)N+k(Nk)c(k+1)2
is ample.
Qualitatively, the λth Schur power of the cotangent bundle of a complete intersection of
at least N/(k+ 1) generic hypersurfaces with large enough degrees will always be ample.
This codimension condition is optimal. Indeed, let us recall the following vanishing
1
July 23, 2019 2
theorem of Brückmann and Rackwitz for M=PN. (For a partition λ, the conjugate
partition λis defined by λ
j:= #{i|λij}.)
Theorem ([BR90]).Let Xbe a smooth complete intersection of codimension cin PN.
Let λbe a partition such that (λ
1+···+λ
c)<(Nc). Then one has the following
vanishing result:
H0(X, SλX) = 0.
Now, assume λsatisfies (k+ 1)c < N (recall k=λ
1). On the one hand, the condition
of the previous theorem for the partition ,mc, becomes
k+···+k
|{z }
×c
< N c,
which is satisfied, hence H0(X, S X) = 0 for mc. On the other hand, the ampleness
of SλXwould imply that SXis globally generated for mlarge enough, by Corollary
1.7 below and Bott’s formula (cf. 1.2). Acccordingly, SλXcannot be ample.
Examples. (1) For k= 1 (symmetric powers), one recovers Xie–Brotbek–Darondeau’s
theorem.
(2) For λ= (1,1) and N= 5, a complete intersection Xof two generic hypersurfaces
of degrees 6.1016 has V2Xample, whereas it cannot have Xample (since
2c= 4 <5).
Let us now briefly sketch the proof of our main theorem. The key ingredient in order
to prove the ampleness of Schur powers SλX(for Xgeneric complete intersection of
codimension c) is to use explicit extrinsic equations to construct morphisms Ψfrom (total
spaces of) flag bundles of the relative tangent space of the universal family X Sof
well-chosen particular complete intersections to simpler generically finite families Y G
of subschemes in some projective spaces (see Diagram 5). The parameter space of Yhas
a natural ample line bundle OG(1). On the one hand, Nakayame’s theorem allows us to
control the augmented base locus of the pullback of OG(1) to Y. On the other hand, by
the very construction of the morphism Ψ, global sections of ΨOG(m)are global sections
of ScmλX(See 2.3). Combining carefully this two facts, one is able to prove the sought
ampleness result.
The paper is organized as follows.
Section 1 is devoted to briefly introducing Schur bundles and flag bundles, and explain-
ing how Hartshorne’s point of view on ampleness can be generalized to this setting. In this
natural framework, substantial difficulties (namely plethysm phenomena and obtaining
extrinsic equations) that would occur following the most naive approach disappear.
Section 2 gives a more detailed overview of the proof sketched above.
Section 3 forms the technical core of the paper. We prove in full detail preliminary
results leading to the construction of the morphisms Ψin Section 4.
In Section 4, we implement the strategy described above in order to complete the
proof.
In Section 5, as a corollary, an application in p-infinitesimal hyperbolicity is given, and
we prove the following.
July 23, 2019 3
Corollary. Let Mbe a projective variety of dimension Nover Cequipped with fixed a
very ample line bundle OM(1). A generic hypersurface X:=H1...Hcof codimension
c,c(k+1) N, with Hi∈ |OM(di)|of degree di1+2c(k+1)(N+k(Nk))c(k+1)+12
is k-infinitesimally hyperbolic.
This property has a geometric interpretation in term of holomorphic maps f:C×
Bp1X: it implies that every such map must be everywhere degenerate, in the sense
that the Jacobian matrix of fis nowhere of maximal rank (Cf. Section 5). For p= 1,
one recovers the notion of hyperbolicity in the sense of Brody.
1. Ampleness, Schur bundles and flag bundles
1.1. Ampleness. For the theory of ample line bundles, we refer to [Laz04]. We briefly
recall here Hartshorne’s point of view on ampleness of vector bundles. A vector bundle E
over a variety Mis said ample if the (dual) tautological line bundle OP(E)(1) is ample
on P(E), the projective bundle of lines in E. Let π:P(E)Mbe the natural
projection. By Bott’s formula [Bot57], one has that for m1
π(OP(E)(m)) = SmE, (1)
where Smis the mth symmetric power. Accordingly, the ampleness of Eis equivalent to
the ampleness of the symmetric powers SmE,m1. As we will see in the sequel, this
point of view can be generalized to Schur powers, looking at appropriate associated flag
bundles.
1.2. Schur bundles and flag bundles. We introduce the terminology and some facts
about partitions, flag varieties, flag bundles, and Schur bundles.
Any partition λ= (λ1 · · · λk>0) with kparts is associated to a Young diagram
of shape λ, which is a collection of cells arranged in left-justified rows: the first row
contains λ1cells, the second λ2, and so on (cf. Figure 1 for an example).
(λ1= 5)
(λ2= 3)
(λ3= 3)
(λ4= 1) s1
s2
s3
Figure 1. Young diagram of the partition λ= (5,32,1), with associated
jump sequence s= (4,3,1).
By construction, if we read the Young diagram from top to bottom via the rows,
we recover the partition λ; observe that if instead we read it from left to right via the
columns, we recover the conjugate partition λmentionned in the introduction. There
is a jump sequence s= (s1>··· > st)associated to a partition λ, defined by:
λi> λi+1 (is)
(where by convention λk+1 = 0). Equivalently, s1>··· > stare the (pairwise distinct)
lengths of the parts of the conjugate partition λ(cf. Figure 1).
July 23, 2019 4
Let Vbe a k-vector space of dimension n. For a decreasing sequence
s= (s1> s2>···> st),
with t1, and s1n1, the partial flag variety FlagsVof sequence sis defined as
FlagsV:={V)F1)··· )Ft)(0) |Fiis a k-vector subspace of dimension si}.
On FlagsV, there is a flag bundle filtration of the trivial bundle FlagsV×Vby universal
subbundles Ust · · · Us1, given over the point η= (F1)... )Ft)FlagsVby
Usi(η) = Fi, for 1is.
Example 1.1. If we take the sequence s= (k), where k1, then FlagsVis the
Grassmannian of k-dimensional subspaces of Vthat we will denote Grass(k, V ), together
with the tautological subbundle Uk.
Line bundles LλVon FlagsVare naturally associated to partitions λwith jump se-
quence sby setting:
LλV:=
t
O
i=1
det(Ui/Ui+1)λsi,
where by convention Ut+1 = 0.
Example 1.2. For the partition λ= (mk),LλVis the mth power of the Plücker line
bundle det(Uk)on Grass(k, V ), that allows to embedd Grass(k, V )in the projective space
P(VkV).
Let us now take Ea vector bundle of rank nover a variety Mand λa partition
with jump sequence s= (s1> s2>··· > st). The previous considerations make sense
fiberwise, and we can glue them via a bundle structure to define:
the projective bundle π: FlagsEX, such that the fiber over xMis
FlagsE
x.
the line bundle LλEover FlagsE, such that (LλE)|FlagsE
x=LλE
x.
Definition 1.3. The Schur bundle associated to a partition λis the direct image
SλE:=π(LλE).
This, together with the vanishing of Riπ(LλE)for i1, is a proposition known as
Bott’s formulas in the literature (Cf. [Bot57]), but we take it as a definition.
Observe that for λ= (m)one recovers (1), i.e. S(m)E=SmE. More generally, SλE
has a standard description as a quotient of products of symmetric powers, and we will
now briefly sketch how to recover it from Definition 1.3.
Proposition 1.4 ([Wey03, Sect. 2.1,Theo. 4.1.8]).The Schur power SλEis a quotient
of
Sλs1(
s1
^E)S(λs2λs1)(
s2
^E)⊗ · · · S(λstλst1)(
st
^E).
Proof. The partial flag bundle Flags(E)has a natural structure of projective bundle:
the product of Plücker embeddings
i: FlagsEP(
s1
^E)×...×P(
st
^E)
July 23, 2019 5
is fiberwise (say along xM) injective, with image a closed algebraic subset of P(Vs1Ex)×
...×P(VstEx). This product of projective varieties is itself projective via the Segre em-
bedding.
Observe that LλEcan be rewritten:
LλE= det(Us1)λs1det(Us2)λs2λs1⊗ · · · det(Ust)λstλst1,
where for 1it,Usiis the relative version of the tautological subbundles introduced
above. Now, by the very definition of the embedding i, this line bundle is actually nothing
but the pullback bundle
LλE=i(OP(Vs1E)(λs1)OP(Vs2E)(λs2λs1)...OP(VstE)(λstλst1)),
where denotes the external tensor product1. Since iis a closed immersion, there is a
natural surjective map
OP(Vs1E)(λs1)OP(Vs2E)(λs2λs1)...OP(VstE)(λstλst1)։iLλE.
(2)
Let p:OP(Vs1E)(b1)...OP(VstE)(bt)Mbe the natural projection. Noticing
that pi=π, and applying pto (2) yields a map
Sλs1(
s1
^E)S(λs2λs1)(
s2
^E)⊗ · · · S(λstλst1)(
st
^E)SλE.
This map is still surjective (See e.g. the proof of Bott’s theorem in [Wey03]), and its
kernel constitutes the so-called exchange relations. One recovers the familiar description
of the Schur power SλEas a quotient of
Sλs1(
s1
^E)S(λs2λs1)(
s2
^E)⊗ · · · S(λstλst1)(
st
^E).
1.3. Ampleness of Schur bundles. There is a natural generalization of the point of
view of Hartshorne to study the ampleness of Schur powers of vector bundle. In [Dem88],
Demailly proved that if Eis ample on M, then LλEis ample on FlagsE. The
statement can be improved by weakening the hypothesis on E: it is enough to suppose
SλEample to deduce the ampleness of LλE. Darondeau asked wether the converse is
true, namely, does the ampleness of the line bundle LλEimply the ampleness of the
vector bundle SλE? In [LN18], the authors answered the question:
Proposition 1.5. ([LN18]) Let λbe a partition with jump sequence s, and let Ea vector
bundle over M. The Schur bundle SλEis ample on Mif and only if the line bundle
LλEis ample on FlagsE.
Example 1.6. In the case λ= (1k), the proposition states that VkEis ample on Mif
and only if the Plücker line bundle is ample on Grass(k, E).
As a corollary, one has the following:
1Let X, Y be varieties, and let p1:X×YX,p2:X×YYbe the first and second projection.
If E1is a vector bundle over X, and E2is a vector bundle over Y, we denote E1E2X×Ythe
vector bundle
E1E2:
=pr
1E1pr
2E2.
This definition generalizes to an arbitrary finite product of varieties.
July 23, 2019 6
Corollary 1.7. Let λbe a partition with jump sequence s, and Ea vector bundle over
M. Then SλEis ample if and only SEis very ample for some m > 0. Accordingly,
SλEis ample if and only if SpλEis ample for any (some) p1.
Proof. The previous proposition allows us to work with the line bundle LλE. Noticing
that (LλE)m=LE, the very definition of SEas the push-forward by πof
LEyields the result.
1.4. Extrinsic equation of Schur powers of cotangent bundles of projective
complete intersection. The first obstacle one encounters in trying to generalize the
method used in [BD18] to the case of Schur powers is in writing extrinsic equations defin-
ing the Schur power of the cotangent bundle of a complete intersection inside the Schur
power of the cotangent bundle of the ambiant space. To illustrate this, let H:=V(P)
be a smooth hypersurface inside M:=PN, N 1, with Pan homogenous polynomial of
degree d. Let Vbe a trivializing open set of OM(1). The local equations of T H inside
T M on Vare as follows:
T H|V={(x, v)T V |P(x) = 0, d(P)V(x, v) = 0},
where (P)Vis the writing of Pon the trivializing open set V. We see that the total
space T H is of codimension 2inside T M , and that we have two natural scalar equations
defining it.
Let us continue this illustration with the case of the kth exterior power. There is a
natural vectorial equation which, added to the equation P= 0, defines locally VkT H
inside VkT M . To see this, let us consider the linear map given on pure tensor by
dE :
VkT M|VVk1T M|V
(x, v1... vk)7−
k
P
j=1
(1)jd(E)V(x, vj)j
v1... vk.
This is well defined since by construction this is the map induced by a k-linear alter-
nating form on T M ×k
|V. If we look at the fiber of xV, one can show that dE(x, .)
is surjective, and since its kernel contains VkTxH, for dimension reason, it is actually
equal to VkTxH. If one restrict oneself to (non-zero) pure tensors (x, v1... vk), the
vectorial equation splits into kscalar equations: d(E)V(x, v1) = ... =d(E)V(x, vk) = 0.
Restricting oneself to non-zero pure tensors is here equivalent to considering the grass-
manian bundle Grass(k, T M ) = Flag(k)T M , and the previous scalar equations define
locally Grass(k, T H )inside Grass(k, T M ):
Grass(k, T H )loc
={(x, [v1... vk]) Grass(k, T M )|E(x) = 0, d(E)V(x, v1) = ···=
d(E)V(x, vk) = 0}.
Working with the appropriate flag variety, which is here the Grassmanian, is hence also
advantageous in this perspective.
Let us finish the illustration with the case of the Schur power associated to a partition
λwith jump sequence s= (s1> ... > st). With analogy to the case of exterior powers,
one is interested in the equations defining locally Flags(T H )inside Flags(T M ), which
are as follows:
July 23, 2019 7
Flags(T H )loc
={(x, F1)... )Ft)Flags(T M )|E(x) = 0, d(E)V(x, v) =
0for all vF1}.
Following this point of view, we will generalize the method followed in [BD18], and
obtain some ampleness results concerning the line bundle LλTHover Flags(TH), where
H:=H1... Hcis a generic complete intersection of codimension c, with some specific
constraints on the codimension and the largeness of the degrees of the hypersurfaces.
With the Proposition 1.5, the ampleness of this line bundle is equivalent to the ampleness
of the vector bundle SλHover H.
1.5. For the rest of the paper, we fix kan algebraically closed field of characteristic 0,
Ma projective algebraic variety of dimension N, equipped with a fixed very ample line
bundle OM(1), and ξ0,...,ξNN+ 1 sections of OM(1) in general position, i.e. each
Di:={ξi= 0}is smooth, and the divisor D=P
i
Diis simple normal crossing. In the
case M=PN, it means that ξ0,...,ξNare homogenous coordinates in PN. We will
adopt the following multi-indexes notations. For a subset J= (j0,...,jN)NN+1, the
support of Jis defined as
[J]:={i[|0, N |]|ji6= 0}
and the length of J is defined as |J|:=j0+···+jN. We will aswell denote
ξJ:=ξj0
0. . . ξjN
NH0(M, OM(|J|)).
Finally, we fix λ= (λ1...,λk>0) a partition with kparts and jump sequence
s= (s1> s2>··· > st), with s1(N1) and t1, whose dual partition writes
λ= (sb1
1,...,sbt
t),biN1. As we are concerned with ampleness of certain vector
bundles SλE, Corollary 1.7 allows us to suppose that gcd(λ1,...,λk) = 1. With the
example 1.1 in mind, we will note k=s1. Note the important role of kin the numerical
hypothesis of our main theorem.
2. Outline of the proof
We assume for simplicity until the end of this section that M=PN.
2.1. Local equations of Flags(TX/S).We consider complete intersections H1...Hc
of codimension c, where each Hpis a hypersurface defined by an homogenous polynomial
of the form
Ep(ap, .):=X
|J|=δp
ap
Jξ(r+1)J,(3)
where J= (j0,...,jN)NN+1,δp>0,ap
JH0(PN,OPN(εp)) with εp>0, and rN.
Accordingly, Ep(ap, .)H0(PN,OPN(εp+ (r+ 1)δp)). We denote, for δN,
Nδ= dim H0(PN,OPN(δ)) = #{JNN+1 | |J|=δ}=N+δ
N
the number of parameters defining a degree δhomogenous polynomial in N+ 1 variables,
and consider
SH0(PN,OPN(ε1))Nδ1×...×H0(PN,OPN(εc))Nδc
July 23, 2019 8
the subspace of parameters (a1,...,ac)of equations (E1(a1, .),...,Ec(ac, .)) as above
defining smooth complete intersections. We denote X Sthe universal family of such
complete intersections, defined in S×PNby the universal equations (E1,...,Ec), and
consider the flag bundle of the relative tangent bundle TX/S
πX: Flags(TX/S) X .
As we saw in 1.4, local equations for Flags(TX/S)S×FlagsTPNare given via the
universal equations (E1,...,Ec)and their relative differentials (dPNE1,...,dPNEc):
Flags(TX/S)loc
=(a1,...,ac),(x, F1)···)Ft)S×Flags(TPN)|E1(a1, x) = ··· =
Ec(ac, x) = 0,dPNE1(a1, x, .)|F1=··· =dPNEc(ac, x, .)|F1= 0.
The equations (3) are such that their differentials can formally be written in the same
form; locally on PN, for 1pc, one can write:
Ep(ap, .) = X
|J|=δp
αp
J(ap, .)(ξr)Jand dPNEp(ap, .) =
loc X
|J|=δp
θp
J(ap, .)(ξr)J,(4)
where the notation =
loc emphasizes that it makes senses only locally, after restricting
oneself to an open subset Vthat trivializes OPN(1)|V. Here, the coefficients αp
Jare
homogenous polynomials of degree εp+δp, and the coefficients θp
Jare 1-form on V
depending on the choice of the trivialization for OPN(1)|V(See subsection 3.1).
2.2. The morphism Ψ.After making the change of variable zi=ξr
ifor 0iN, the
equations (4) become
X
|J|=δp
αp
JzJand X
|J|=δp
θp
JzJfor 1pc.
We fix 1pc, and consider this time αp
Jand θp
Jas variables in k. As mentionned
earlier, we implicitly assume that we restrict ourselves to a trivializing open set Vof
OPN(1). One can then form, for apSp:=H0(PN,OPN(εp))Nδp,xVand v1,...,vk
TxPN, the following matrix
Ap(ap; (x;v1,...,vk)) =
((αp
J)(ap, x))J
(θp
J(ap, x, v1))J
.
.
.
(θp
J(ap, x, vk))J
,
which is of size (k+ 1) ×Nδp. Let us suppose that Ap(ap; (x;v1,...,vk)) is of maximal
rank, namely k+ 1 (this condition does not depend on the trivialization chosen to write
the matrix). In this case, η= (F1)... )Ft), where Fiis the subvector space of kNδp
spanned by the si+ 1 first rows of the matrix, defines an element of Flags+kNδp, where
s+= (s1+ 1 > ... > st+ 1). One can then consider the coordinates of ηFlags+kNδp,
which are given in term of some minors of the matrix. Now, one would like to be able
to make the previous considerations on the whole space PN, for apbelonging to an open
set of the space of parameters: one can show that it is possible as soon as δpis taken
July 23, 2019 9
large enough, and it is the object of 3.3. We denote S
pSpsuch an open set. Let us
mention that here lies the reason why k=s1plays a particular role: the condition so as
to make the previous reasoning depends on konly, and not on the whole partition λ.
Suppose apis in S
p. One can show (cf. 3.2) that the right kind of product of coordinates
of ηcan be interpreted as a global section of LλTPN, tensorized by a power of π
PNOPN(1),
where πPN: FlagsTPNPNis the natural projection. Considering every section arising
in this way allows one to define a morphism ϕp(ap, .) : FlagsTPNPM. Here Mis a
certain natural number. This morphism depends algebraically on S
p, so one can see it
as a morphism
ϕp:S
p×FlagsTPNPM.
Let us be more specific on the image of the morphism: by construction, ϕptakes values
in embedδp(Flags+kNδp):=Gδp, where
embedδp:
Flags+kNδpGrass(s1+ 1,kNδp)b1×...×Grass(st+ 1,kNδp)bt
(∆1)2)···)t)7−(∆1,...,1,2,...,2,...,t,...,t)
and accordingly:
ϕp:S
p×FlagsTPNGδp.
Define S:=S
1×...×S
cand G:=Gδ1×...×Gδc, and construct the morphism
Ψ:
S×FlagsTPNG×PN
(a, η)7−(ϕ1(a1, η),...p(ac, η)),[ξr
0(πPN(η)) : ... :ξr
N(πPN(η))]
where a= (a1,...,ac). By construction, Ψ(Flags(TX/S)) factors through YG×PN
defined as:
Y:=(∆1
1,...,1
t)×...×(∆c
1,...,c
t), zG×PN| ∀ 1ic, i
1(z) = 0
where «i
1(z) = 0» means that for every vector v= (vJ)|J|=δii
1,P
|J|=Nδi
vJzJ= 0.
To sum up, we obtain the following diagram:
Flags(TX/S)Ψ//
Y//
ρ
PN
SG
(5)
where ρis the first projection.
2.3. The model situation. The product G=Gδ1×. . . Gδcis naturally endowed with
an ample line bundle OG((1,...,1)) = OGδ1(1) ... OGδc(1). Denote OX(1) :=
(OSOPN(1))|X ; by construction of the morphism Ψ, the pullback by Ψof (OG(m)
OPN(1))|Y, where m= (m1,...,mc)Nc, is of the form
Ψ((OG(m)OPN(1))|Y) = LλTX⊗|m|
/Sπ
XOX(r+C(λ, ε,δ,m)),(6)
where C(λ, ε,δ,m)is a constant that depends on the partition λ, the c-uples ε=
(ε1,...,εc),δ= (δ1,...,δc)and m= (m1,...,mc).
Suppose now that c(k+ 1) Nso that the morphism ρis generically finite. Ac-
cordingly, the pullback ρOG((1,...,1)) of the ample line bundle OG((1,...,1)) is big
July 23, 2019 10
and nef. One can obtain a geometric control of the augmented base locus Bs((OG(m)
OPN(1))|Y)for mwith large enough components via Nakayame’s theorem, but it does
not provide an explicit bound. Instead, one can use an «effective Nakayame’s theorem
»([Den16], and cf. 4.3.1) in order to obtain that for mwith large enough components
(the bound depending on δ1,...,δc):
Bs((OG(m)OPN(1))|Y)Exc(ρ)(7)
where Exc(ρ)is the reunion of all positive dimensional fibers of the morphism ρ:YG.
2.4. Pulling back the positivity from Yto Flags(TX/S).Suppose m= (m1,...,mc)
have large enough components so that (7) is valid. The next and final step is to investi-
gate the base locus of the pull-back by Ψof OG(m)OPN(1))|Y. This is the object
of 4.2, which shows in particular that for δlarge enough and for a general ain S, any
curve C ⊂ FlagsT Hasatisfies Ψ(C)6⊂ Exc(ρ), and a fortiori with (7) satisfies
Ψ(C)6⊂ Bs((OG(m)OPN(1))|Y).(8)
Let us denote
Qa:=LλT H |m|
aπ
aOHa(r+C(λ, ε,δ,m)) (9)
where abelongs to S, and πa: Flags(T Ha)Hais the natural projection. In view
of (6) and (8), any curve C ⊂ FlagsT Hasatisfies
Qa· C 0.
This shows that Qais nef. With (9), if one picks r > C(λ, ε,δ,m),LλT H ⊗|m|
awrites
as the sum of a nef line bundle with the pull-back by π
aof an ample line bundle. As
LλT H ⊗|m|
ais ample along every fiber of πa(since it restricts to LλTxH⊗|m|
aif we look
at the fiber over xM), we deduce that LλT H ⊗|m|
ais ample, and so is LλT Ha. By the
openess property of ampleness, and the Proposition 1.5, it implies the main theorem.
However, as we will see in the sequel during the proof, technical difficulties prevent us
from applying directly what was said above. Instead of working globally, we will be forced
to work on the stratification of Flags(TPN)induced by the hyperplane coordinates.
3. Notations and preliminary constructions
3.1. Notations. Let ε1,δ1,r1, and S:=H0(M, OM(ε))Nδ, where Nδ=
N+δ
N. For a= (aJ)JNN+1,|J|=δS, let us form the following bihomogenous section of
OM(ε+ (r+ 1)δ)
E(a, .): x7→ X
|J|=δ
aJ(x)ξ(x)(r+1)J,
where we recall the multi-indexes notations ξ(x)J=ξ0(x)j0. . . ξN(x)jNwhere J=
(j0,...,jN)NN+1. Let us write E(a, .)under the following form
E(a, .) = X
|J|=δ
[aJξJ]ξrJ
=X
|J|=δ
αJ(a, .)ξrJ
July 23, 2019 11
where αJ(a, .) = aJξJH0(M, OM(ε+δ)). We can then write, formally, that
d(E(a, .)) = X
|J|=δ
θJ(a, .)ξrJ ,
where θJ(a, .) = ξJdaJ+ (r+ 1)aJdξJ. Let us insist on the fact that the previous
equality is formal: it makes sense only when writing the sections on a trivializing open
set of OM(1).
Recall that we denote πM: Flags(T M )Mthe projection on M. We will sometimes
write, for sake of notation, s(η)for s(πM(η)), where ηFlags(T M )and sis a section
of a line bundle on M.
3.2. Definition of a preliminary morphism. Let us denote, for ain the space of pa-
rameters S,Hathe smooth hypersurface in Mdefined by the equation (E(a, .) = 0). In
the spirit of [BD18], the first step is to define a preliminary morphism ∆(a, .) : Flags(T M )
G(Gwill be defined later), that depends algebraically on a, which
keeps track of the equations defining Flags(T Ha)inside Flags(T M )by its very
definition.
turns out to be a morphism for aSS, where Sis non-empty open set of
S.
is such that Gpossesses an ample line bundle whose pull-back by ∆(a, .)gives
LλT M tensorized by a power of π
MOM(1).
Let Vbe an open set that trivializes OM(1). Let us consider, for xV, the following
map from TxMto kNδ
fx:TxMkNδ, v 7→ (θJ(a, x, v))J,
where we implicitly assume that θJis written in the trivialization chosen.
For µ= (µ1,...,µs)a partition, we will denote µ+= (µ1+ 1,...,µs+ 1). Let us form
the following map Fxfrom Flags(TxM)to Flags+kNδ
Fx: Flags(TxM)Flags+kNδ,(F1)... )Ft)7→ (fxF1+kv)... )fxFt+kv),
where v= (αJ(a, x))JkNδ. This is well defined only if for every free family (v1,...,vk)
of TxM,((αJ(a, x))J, fxv1,...,fxvk)remains a free family of kNδ. Equivalently, it is well
defined if the following (k+ 1) ×Nδmatrix
A(a; (x;v1,...,vk)) =
((αJ)(a, x))J
(θJ(a, x, v1))J
.
.
.
(θJ(a, x, vk))J
is of maximal rank for each free family (v1,...vk)TxM. Let us then consider the
following condition
()xM, (v1,...,vk)free family in TxM, A(a; (x, v1,...,vk)) is of maximal rank.
Observe that it makes sense since the rank does not depend on the trivialization chosen
to form the matrix: any other matrix formed with another trivialization would have the
July 23, 2019 12
same rank. As we will be interested in coordinates in Flags+kNδ, we make the following
definition:
Definition 3.1. Let k, N be natural numbers, and let A∈ M(k+1)×N(k)be a matrix
of rank k+ 1. We will call Plücker coordinates of size lk+ 1 the determinant of a
matrix obtained from Aby keeping the lfirst rows, and keeping ldistinct columns.
Let us now state a lemma that will allow us to define the wanted preliminary morphism:
Lemma 3.2. Let Lbe a line bundle on M. Let l2, and let ω1,...,ωlbe global sections
of L. The following map:
l1
^T M k,(x, v1... vl1)7→ det
ω1(x). . . ωl(x)
1(x, v1). . . l(x, v1)
. .
. .
. .
1(x, vl1). . . l(x, vl1)
defines a global section of (Vl1T M )Ll.
Proof. For Van open set that trivializes the line bundle L, and ωa global section of L,
we will denote (ω)Vthe representation of the section ωon the trivializing open set V.
Let us fix Vand Vtwo open sets of Mthat trivializes the line bundle L, and let us
denote gV V the transition map. Let us fix xVV. We have that, on VV, for
every 1il1:
(ωi)V=gV V (ωi)Vand d(ωi)V=gV V d(ωi)V+ (ωi)VdgV V .
Let us denote, for v1,...,vl1TxMl1,
BV(x;v1,...,vl1) =
(ω1)V(x)... (ωl)V(x)
d(ω1)V(x, v1). . . d(ωl)V(x, v1)
. .
. .
. .
d(ω1)V(x, vl1). . . d(ωl)V(x, vl1)
.
We prove by induction on l2that, for every v1,...,vl1TxMl1, the following
equality is satisfied:
det BV(x;v1,...,vl1)=gl
V V det BV(x;v1,...,vl1).(10)
For l= 2, it follows from a straightforward computation. Fix l > 2, and suppose the
equality (10) true for the previous ranks. Developping det BV(x;v1,...,vl1)according
to the last row and using the induction hypothesis, we get:
det BV(x;v1,...,vl1)=
l
X
i=1
(1)l+id(ωi)V(x, vl1)gl1
V V det B(l,i)
V(x;v1,...,vl1),
July 23, 2019 13
where B(l,i)
Vis the matrix obtained from BVby suppressing the lth row and the ith
column. The previous equality can be written as
det BV(x;v1,...,vl1)=gl1
V V det
(ω1)V(x)... (ωl)V(x)
d(ω1)V(x, v1). . . d(ωl)V(x, v1)
. .
. .
. .
d(ω1)V(x, vl2). . . d(ωl)V(x, vl2)
d(ω1)V(x, vl1). . . d(ωl)V(x, vl1)
.
Using the determinant by blocks formula, the right side of the equality writes:
X
i<j
(1)l+i+jgl1
V V det (ωi)V(x) (ωj)V(x)
d(ωi)V(x, vl1)d(ωj)V(x, vl1)det B(1,i),(l,j)
V(x;v1,...,vl1)
where B(1,i),(l,j)
Vis the matrix obtained from BVby suppressing the 1st and ith row
aswell as the ith and the jth column. It rewrites
X
i<j
(1)l+i+jgl2
V V det (ωi)V(x) (ωj)V(x)
d(ωi)V(x, vl1)d(ωj)V(x, vl1)det B(1,i),(l,j)
V(x;v1,...,vl1),
Using the case l= 2, and the reverse of the previous determinant by blocks formula
yields the equation (10), and concludes the induction.
Now,
TxMl1k,(v1,...,vl1)7→ det BV(x;v1,...,vl1)
is a (l1)-linear alternating form, so it induces a linear map from Vl1TxMto k. From
this, we infer that
l1
^T M|Vk,(x, v1... vl1)7→ det BV(x;v1,...,vl1)
defines a section of (Vl1T M )
|V. And the previous calculation shows exactly that those
sections glue together to give a global section of (Vl1T M )Ll.
As an immediate application of this lemma, we have the following two corollaries:
Corollary 3.3. A Plücker coordinate of size lk+ 1 of the matrix A(a; (x;v1,...,vk))
defines a global section of (Vl1T M )⊗ OM(l(ε+δ)).
Proof. Apply the previous lemma to ldistinct sections (aJiξ(r+1)Ji)1ilH0(M, OM(ε+
(r+1)δ)), and write (B)Vthe matrix as in the proof of the previous lemma. If we denote
(P)Vthe matrix obtained from (A)V(a;.)by keeping the columns corresponding to the
indexes (Ji)1il, observe that
(B)V= (ξrJ )V(P)V,
which implies that det(B)V= (ξlrJ )Vdet(P)V. From this, we deduce as in the previous
lemma that the local sections det(P)Vof Vl1T M
|Vglue together to give a global section
of Vl1T M ⊗ OM(l(ε+ (r+ 1)δ)lrδ) = Vl1T M ⊗ OM(l(ε+δ)).
July 23, 2019 14
Corollary 3.4. Let p1
1,...,pb1
1,...,p1
t,...,pbt
tbe a collection Plücker coordinates of size
respectively s1+ 1,...,st+ 1 of the matrix
A(a; (x;v1,...,vk)) =
((αJ)(a, x))J
(θJ(a, x, v1))J
.
.
.
(θJ(a, x, vk))J
.
Then p1
1...pb1
1... p1
t... pbt
tdefines a global section of [(Vs1T M )b1...
(VstT M )bt]⊗ OM(|λ
+|(ε+δ)).
Keeping the notation of the previous corollary, the section p1
1...pb1
1...p1
t...pbt
t
induces a section of LλT M π
MOM(|λ
+|(ε+δ)) (See e.g. Propostion 1.4).
Let us suppose that ais such that the condition ()holds, and let us consider all
the sections of LλT M π
MOM(|λ
+|(ε+δ)) we can create in the way just described.
The fact that ()is satisfied implies that the sections cannot vanish simultaneously on
a given point of Flags(T M ), and accordingly we can create with them in the usual way
a morphism
∆(a, .): Flags(T M )Grass(s1+ 1,kNδ)b1×...×Grass(st+ 1,kNδ)bt.
Note here that we identify Grass(s1+ 1,kNδ)b1×...×Grass(st+ 1,kNδ)btwith its image
by the Segre embedding σ.
Observe that the image of ∆(a, .)can in fact factor through Gδ:= embedδ(Flags+kNδ),
where embedδis the embedding of Flags+kNδinside Grass(s1+1,kNδ)b1×...×Grass(st+
1,kNδ)btgiven by
embedδ: (∆1)2)... )t)7→ (∆1,...,1,2,...,2,...,t,...,t).
Indeed, let us fix η= (x, F1)... )Ft)an element of FlagsT M , and let us consider
the section sobtained via p1
1... pb1
1... p1
t... pbt
t. By construction, s(η)
is a coordinate of embedδ(Fx(η)) in Grass(s1+ 1,kNδ)b1×... ×Grass(st+ 1,kNδ)bt.
Considering all the sections of the previous form is equivalent to considering all the
coordinates of embedδ(Fx(η)) in Grass(s1+ 1,kNδ)b1×...×Grass(st+ 1,kNδ)bt, hence
the factorization.
For u= (u1
1,...,ub1
1,...,u1
t,...,ubt
t)Nb1+...+bt, define the following line bundles
over Gδ:
Qδ(u):=OGrass(s1+1,kNδ)(u1
1)...OGrass(s1+1,kNδ)(uc
1)...OGrass(st+1,kNδ)(ut
1)
...OGrass(st+1,kNδ)(ut
t)|Gδ.
Note that for u= (1,...,1),Qδ(u)is the canonical line bundle on Gδ.
We can now state the following lemma, which follows immediately from the construc-
tion of ∆(a, .):
Lemma 3.5. For athat satisfies (), one has
∆(a, .)Qδ((1,...,1)) = LλT M π
MOM(|λ
+|(ε+δ)).
July 23, 2019 15
To see how we keep track of the equations of Flags(T Ha)inside Flags(T M ), let us
first make the following definition:
Definition 3.6. Let be a s1+ 1-space in Grass(s1+ 1,kNδ). If v= (vJ)|J|=Nδis a
vector inside this space, we form the polynomial P,v(z):=P
|J|=Nδ
vJzJ,zPN. We will
write «∆(z) = 0» to mean that P,v(z) = 0 for every v6= 0 .
Let us then form the following morphism
Ψ(a, .): Flags(T M )Gδ×PN, η 7→ (∆(a, η),[ξ(η)r]),
where [ξ(η)r]stands for [ξ0(η)r:... :ξN(η)r]. What was meant by keeping track of the
equations of Flags(T Ha)inside Flags(T M )is that, by construction, Ψ(a,FlagsT Ha)is
included in the following set:
Yδ:={((∆1,...,1,...,t,...,t), z)Gδ×PN|1(z) = 0}.
Let
gδ:
GδFlags+kNδ
(∆1,...,1,2,...,2,...,t,...,t)7−(∆1)2)... )t)
be the inverse of embedδ, and let
Zδ:= (gδ×IdPN)(Yδ) = {((∆1)... )t), z)Flags+kNδ×PN|1(z) = 0}.
Let us define aswell
pδ: Flags+kNδGrass(s1+ 1,kNδ)
the natural projection onto Grass(s1+ 1,kNδ), and
Y
δ:= (pδ×IdPN)(Zδ) = {(∆, z)Grass(s1+ 1,kNδ)×PN|∆(z) = 0}.
We then have the following commutative diagram
Yδ
gδ×IdPN//
ρ
Zδ
pδ×IdPN//
Y
δ
ρ
Gδ
gδ//Flags+kNδpδ//Grass(k+ 1,kNδ),
(11)
where the vertical arrows are the natural projections.
The next part will be devoted to the existence of an open subset SSon which ()
is satisfied, allowing us to make all the previous considerations. Before going on, let us
make an observation that we will need later (3.3 and 4.2.2). If we consider the particular
case where λ= (1k), the condition ()is the same as when one considers a partition λ
with jump sequence s= (s1=k > ··· > st), which is straightforward by definition. We
denote for asatisfying ()
Ψ(a, .): Grass(k, T M )G
δ×PN
July 23, 2019 16
the morphism constructed in this particular case, where G
δ= Grass(k+1, T M ). Observe
that the following diagram is commutative
Grass(k, T M )
Ψ(a,.)
FlagsT M Ψ(a,.)
//
q
33
Gδ×PN
pδgδ×IdPN
//G
δ×PN
,
(12)
where
q: FlagsT M Grass(k, T M ),(x, F1)··· )Ft)7→ (x, F1),
which is straightforward by the very construction of the morphisms Ψand Ψ.
3.3. Existence of the open subset S.Due to the specific form of the equations cho-
sen, we need to slightly change the condition (). Let us define the following stratification
of M.
Definition 3.7. For I({0,...,N}, denote DI=TiIDiwhere Di={ξi= 0}, aswell
as MI=DI\Ti/IDi.
If |I|=Nk0,dim MI=k0, and the M
Isstratify M.
This stratification induces in a natural way a stratification on FlagsT M by considering
FlagsT M|MI:=π1
M(MI).
Instead of considering (), one instead considers
(, I)xMI,(v1,··· , vk)free family in TxM, A(a; (x, v1,...,vk)) is of maximal rank.
We will prove that there exists an open set Ssuch that (, I)is satisfied for every
I({0,...,N},|I|< N.
3.3.1. Stratification of Grass(k , T M).As we saw at the end of the last subsection, we
can restrict ourselves to the particular case λ= (1k). We first describe a stratification of
Grass(k, T M )that we are going to need in the course of the proof.
Definition 3.8. Let I({0,...,N},|I|=Nk0, and let II,|I|=Nk1, where
k1k. Define:
Σ(I, I) = {η= (x, v1...vk)Grass(k, T M|MI)|iIi(x, v1) = ···i(x, vk) = 0}.
Note that since II, the nullity condition given in the definition does not depend
on the local coordinates chosen to write the differentials i.
Lemma 3.9. Σ(I, I)is of dimension k0+k(k1k).
Proof. With respect to M,Σ(I , I)is locally defined by |I|=Nk0independant equa-
tions. Let us see what happens in the fiber over xM. By definition of Σ(I, I), the
fiber is Grass(k, T
iI
TxDi), where we recall that Di={ξi= 0}. The computation of the
dimension of Grass(k, T
iI
TxDi)yields the result.
July 23, 2019 17
Denote, for I({0,...,N}
P(kI):={(z0,...,zN)PN| ∀iI, zi= 0}.
The inclusion P(kI)֒PNinduces a projection:
resI:L|J|=δkH0(PN,OPN(δ)) H0(P(kI),OP(kI)(δ)) L|J|=δ, [J]I=k
(tJ)|J|=δ7−(tJ)|J|=δ, [J]I=
For J= (j0,...,jN)NN+1, we say that jiis the weight of Jcorresponding to the
ith index.
3.3.2. Study of the rank of a linear map. Let us fix η= (x, v1...vk)Grass(k, T M ),
where (v1,...,vk)is a free family of TxM, and let us study the rank of the following map
ϕη:S(kk+1)Nδ,a= (aJ)J7→ A(a; (x, v1,...,vk)),
where we recall that A(a; (x;v1,...,vk)) =
((αJ)(a, x))J
(θJ(a, x, v1))J
.
.
.
(θJ(a, x, vk))J
. Let us aswell recall that
when writing θJ, we implicitly assumed that we fixed a trivialization of OM(1) around
xM, and wrote θJunder this trivialization.
Proposition 3.10. Let I, I be like in the definitions B.II.3.a, with |I|=Nk0and
|I|=Nk1,k1k.
(i) For ηΣ(I , I),rank(ϕη) = (k+ 1)k0+δ
k0+ (k1k0)k0+δ1
k0.
(ii) Furthermore, (resI×... ×resI)ϕηis a surjective map from Sto
H0(P(kI),OP(kI)(δ))×(k+1).
Proof. The first thing to observe is that the map ϕηcan be represented by blocks
∗ ∗
∗ ∗0... 0
0∗ ∗
∗ ∗... 0
.
. .
. .
0 0 ...∗ ∗
∗ ∗
where each block is a matrix with k+ 1 rows and dim(H0(M, OM(ε))) columns,
and the number of such blocks is equal to Nδ. To see this, one considers Sas
(H0(M, OM(ε)))JNN+1 ,|J|=δand kNδas (kk+1)JNN+1 ,|J|=δ; each vertical entity in the
previous block representation corresponds to the image of {0} × ... ×H0(M , OM(ε)) ×
{0} × ...× {0}, the non zero-term corresponding to a certain index J.
Now, let us study the rank of a block corresponding to a given index J.
1st case: Jsuch that ξJ(η) = 0.For every a, one has that:
July 23, 2019 18
(i) αJ(a, η) = aJ(η)ξJ(η) = 0.
(ii) θJ(a, x, vi) = (r+ 1)aJ(η)dξJ(x, vi)for every 1ik.
Observe that dξJ(x, .)is identically zero as soon as |[J]I| ≥ 2, where we recall that
[J]is the support of the multi-index J. If [J]Iis a single index, observe that if the
weight of Jcorresponding to this index is stricly greather than one, then dξJ(x, .)is still
identically zero. So, the only case where one can hope for a non-zero block on the Jth
spot is when [J]Iis a singleton, with the weight of Jat this singleton equal to one. In
this case, observe that the columns of the block are all proportionnal to
0
dξJ(x, v1)
.
.
.
dξJ(x, vk)
,
so the rank is at most one, and to get a rank equal to one, we need to have at least one
isuch that dξJ(x, vi)6= 0. By hypothesis, we took ηΣ(I , I), so the only case where
this is satisfied is when Jis such that
(i) [J]Iis a singleton, with the weight of Jat this singleton being one.
(ii) The singleton [J]Iis in I\I.
We will, for sake of notation, write abusively «[J]I={1}» to mean these two condi-
tions. One counts that there exists (k1k0)k0+δ1
k0such indexes J.
2nd case: Jsuch that ξJ(η)6= 0.First, observe that under the hypothesis ε1, every
element of (TxM)can be written under the form da(x, .)where aH0(M, OM(ε))
satisfies a(x) = 0. Let asuch that aJ(x) = 0. We have that for every 1ik:
θJ(a, x, vi) = ξJ(η)daJ(x, vi).
From the previous observation, for every 1ik, we can find aisuch that
θJ(ai, x, vj) = δj
i
for every 1jk. And since we can always find aH0(M, OM(ε))\Vect((a1)J,...,(ak)J)
such that a(η)6= 0, we deduce that the rank of the Jth block is at least equal to k+ 1,
and it is thus equal to (k+ 1), since the block has (k+ 1) rows. Since ηΣ(I, I ), the
number of indexes Jsuch that ξJ(η)6= 0 is equal to k0+δ
k0, and for those indexes, the
rank of the corresponding block is thus maximal, i.e. equal to k+ 1.
Combining the first and second case, due to the block decomposition, we deduce that:
rank(ϕη) = (k+ 1)k0+δ
k0+ (k1k0)k0+δ1
k0
which completes the proof of (i).
As for the proof of (ii), if we compose by the projection onto H0(P(kI),OP(kI)(δ))×k+1,
we keep only the block of the second kind, and thus the rank is full.
3.3.3. Existence of the open subset S0.For I⊂ {0, ..., N },|I|< N, let us denote
B(I)loc
={(a, η)S×Grass(k, T M|MI)|rank A(a;η)is stricly less than k+ 1}.
July 23, 2019 19
Let us observe that if we prove that B(I)does not dominate Sunder the first projection
proj1:S×Grass(k , T M)S,
then the condition (, I)is satisfied on a (Zariski) open set S,I S, since (, I )is then
an open condition satisfied by at least one point.
We denote
π
M: Grass(k, T M )M
the projection onto M, aswell as
proj2:S×Grass(k , T M )Grass(k, T M )
and
pr1:S×MS.
Proposition 3.11. For δk+ 1, for every I⊂ {0, ..., N },|I|< N ,B(I)does not
dominate Sunder proj1.
Proof. Observe first that it is enough to prove that B(I)[S×Σ(I, I )] does not dom-
inate S×Σ(I, I)under proj1for every II({0,...,N}since (Σ(I, I ))Istratifies
Grass(k, T MMI).
Let us denote
Z(I, I) = B(I)[S×Σ(I, I )],
where |I|=Nk0and |I|=Nk1, for 1k0k1, and let us decompose Z(I , I)
under the following form
Z(I, I) = Zα(I, I )G[Z(I , I)\Zα(I , I)],
where
Zα(I, I) = {(a, η)S×Σ(I , I)|αJ(a, η) = 0 for every J}.
Like before, it is enough to prove that Zα(I, I )nor Z(I, I )\Zα(I, I)dominates Sunder
proj1.
1st case: Zα(I, I ). Let us denote
Zα(I) = (idS×π
M)(Zα(I, I)) = {(a, η)S×MI|αJ(a, η) = 0 J}.
Observe that proj1(Zα(I, I )) = pr1(Zα(I)), thus Zα(I, I)dominates Sunder proj1if
and only if Zα(I)dominates Sunder pr1. We fix xMI. For any a= (aJ)|J|=δ, one
has
(αJ(a, x) = 0 J)(aJ(x) = 0 Jsuch that [J]I=).
Indeed, αJ(a, x) = aJ(x)ξJ(x), thus αJ(a, x) = 0 if and only if aJ(x) = 0 or ξJ(x) = 0.
Since xMI,ξi(x) = 0 if and only if iI. Thus we have that ξJ(x) = 0 if and
only if [J]I6=, and this gives the equivalence. Since there are k0+δ
k0multi-indexes
Jsatisfying the right-hand side condition, each multi-index imposing an independant
condition, we have that dim(Zα(I)pr1
2(x)) dim Sk0+δ
k0. From this, we deduce
that
dim(Zα(I)) dim(S)dim MI+ max
xMI
dim(Zα(I)pr1
2(x)) dim S
k0k0+δ
k0<0
July 23, 2019 20
as soon as δ2. Thus Zα(I, I )does not dominate S.
2nd case: Z(I , I)\Zα(I , I). Let us fix η= (x, [v1... vk]) Σ(I, I), and let us fix
xVMan open subset that trivializes OM(1). Let us denote
Zη:= (proj2)1(η)[Z(I , I)\Zα(I, I)],
and observe that Zηis isomorphic to proj1(Zη)under proj1. Let us denote, for i1,Vi
the vectorial space that consist of the elements (w0, w1,...,wk)k(k+1)Nδthat satisfy
(1.i)wiVect(w0,...,wi1, wi+1 ,...,wk),
(2) For Jsuch that [J]I6=,
(w0)J
.
.
.
(wk)J
k.
0
dξJ(x, v1)
.
.
.
dξJ(x, vk)
,
and observe that ϕη(proj1(Zη)) is included in Si1Vi. Indeed, by definition of Z(I , I)\
Zα(I, I), the first row of every element of ϕη(proj1(Zη)) is non-zero, and thus the con-
ditions (1.i)1ikexpresses that the rank condition on A(a, η)is not satisfied; as for (2),
it follows from the fact that ηΣ(I, I )(Cf. the discussion of the 1st case in the course
of the proof of Proposition 3.10).
We will now evaluate the dimension of Vi. As we saw in the course of the proof of
proposition 3.10, there are C1= (k1k0)k0+δ1
k0indexes Jsuch that
0
dξJ(x, v1)
.
.
.
dξJ(x, vk)
is
non-zero (they correspond to the indexes Jsuch that «[J]I={1}»: cf. the proof
of Proposition 3.10 for the notation) and C0=k0+δ
k0indexes Jsuch that [J]I=.
Accordingly, the condition (2) allows one to see Vias living in
(M
J,[J]I=
k)×k+1 ×(M
J,[J]I={1}
k)k(k+1)C0+C1.
Let us now use the conditions (1.i) and (2) to evalute the dimension of Vi, and to do so,
let us construct the (algebraic) morphism Li:
kk×(L[J]I=k)×k×L[J]I={1}k(L[J]I=k)×(k+1) ×L[J]I={1}k
(ωj)j6=i,λj
Jj6=i
[J]I=,(λJ)[J]I={1}7→ λj
Jj6=i
[J]I=,P
j6=i
ωjλj
J[J]I=,(λJ)JI={1}
With the condition (1.i), ViIm(Li), and thus Vi=Li(L1
i(Vi)). But one can say more
with (1.i) combined with (2), namely that L1
i(Vi)is included in
n(ωj)j6=i,(λj
J)[J]I=j6=i,(λJ)[J]I={1}| ∀J, J [I] =
{1}, λJ[dξJ(x, vi)P
j6=i
ωjdξJ(x, vj)] = 0o
July 23, 2019 21
This set is a union of subvectorial spaces of codimension C1of kk×(kC0)k×kC1. There-
fore, its dimension (as an algebraic set) is equal to k+kC0. Since Vi=Li(L1
i(Vi)),
dim Vik+kC0, and accordingly one deduces that
dim ϕη(proj1(Zη)) k+kk0+δ
k0
since ϕη(proj1(Zη)) Si1Vi. Using this inequality, with proposition 3.10, one infers
that
dim(Zηproj1(Zη)) k+kk0+δ
k0+ dim(ker(ϕη))
=k+kk0+δ
k0+ dim Srank(ϕη)
kk0+δ
k0(k1k0)k0+δ1
k0+ dim S.
Since this is true for every ηΣ(I, I ), one has that
dim(Z(I, I)\Zα(I, I)) dim S
dim Σ(I, I) + max
ηΣ(I,I )dim Zηdim S
k+k0+k(k1k)k0+δ
k0(k1k0)k0+δ1
k0
= (k+ 1)k0k0+δ
k0+kk2+ (k1k0)kk0+δ1
k0,
which is stricly negative for any k1k01as soon as δk+ 1 (Let us recall that
by hypothesis on I, namely |I|< N,k01). To see this, observe for instance that for
n, N N1,n+N
nnN, with a strict inequality for n, N 2(which can be proved by
recurrence on n+N2).
With this proposition, we deduce that for every I,|I|< N, the condition (, I)
is satisfied on an open set S,I S. Since Sis an irreducible variety (as product of
irreducible varieties), the intersection of these open sets is a non-empty open set, that
we will write S. Following what was done in 3.2 in restriction to MI, one can define for
every aSa morphism
I(a, .): FlagsT M|MIGδ
such that
I(a, .)Qδ((1,...,1)) = LλT M|MIπ
MIOM(|λ
+|(ε+δ)),
where πMI: FlagsT M|MIMIis the projection. Then one constructs the morphism
ΨI(a, .): FlagsT M|MIGδ×P(kI), η 7→ (∆(a, η),[ξ(η)r]),
which satisfies, for asuch that Hais smooth
ΨI(a,Flags(T Ha)MI)⊂ Yδ(I):={((∆1,...,1,...,t,...,t), z)
Gδ×P(kI)|1(z) = 0}.
July 23, 2019 22
Observe that the morphism ΨI(a, .)depends algebraically on aS; accordingly, we
have a morphism
ΨI:S×FlagsT M|MIGδ×P(kI)
4. Ampleness of Schur bundles of cotangent bundles of general
complete intersections
4.1. The morphism Ψ.From now on, we consider not a single hypersurface Ha, but
a complete intersection Ha:=Ha1...Hacof codimension c, where a= (a1,...,ac)
and aiis taken in the space of parameters Si:=H0(M, OM(εi)))JNN+1,|J|=δi, where
εiN1, and δiis large enough as in the Proposition 3.11. We denote
X:={(a1,...,ac;x)S1×...×Sc×M|E1(a1, x) = ··· =Ec(ac, x) = 0}
the universal family of such complete intersection, and we will denote, for each 1ic,
S
ithe open set on which (, I)is valid for every I⊂ {0,...,N},|I|< N . According to
the last section, for each 1ic, one can define
i
I:S
i×FlagsT M|MIGδi.
The following lemma, which is for instance proved in [BD18]
Lemma 4.1. A general fiber of the family X S1×...×Scis smooth.
allows us to restrict ourselves to the open dense subset SS1×...×Scparametrizing
smooth complete intersection varieties, and we denote X Sthe corresponding universal
family.
Denote S:=S(S
1× · · · × S
c), and form the following morphism
ΨI:S×FlagsT M|MIG×P(kI),(a, η)7→ 1
I(a1, η)×...×c
I(ac, η)×[ξ(η)r]
where G:=Gδ1×...×Gδc. As in the end of subsection 3.2, we define
Y:={((∆1
1,...,1
t)×...×(∆c
1,...,c
t), z)G×PN| ∀1ic, i
1(z) = 0}
as well as
gδ:=gδ1×...×gδcand Z:= (gδ×IdPN)(Y)
and
pδ:=pδ1×...×pδcand Y:= (pδ×IdPN)(Z).
We have the following commutative diagram as in (11):
Ygδ×IdPN//
ρ
Zpδ×IdPN//
Y
ρ
Ggδ//
c
Q
i=1
Flags+kNδipδ//
c
Q
i=1
Grass(k+ 1,kNδi):=G
(13)
where the vertical arrows are the natural projections. We denote for I({0,··· , N }
Y(I):=YG×P(kI),
and similarly for Z(I)and Y(I).
July 23, 2019 23
From the definition of the morphism, observe that for aS
ΨI(a,Flags(T Ha)|MI)Y(I).
Accordingly, we have the following commutative diagram as in (5):
Flags(TX/S)ΨI//
Y(I)//
ρI
P(kI)
SG
(14)
where ρIis the restriction of the first projection ρto Y(I). Suppose that c(k+ 1) N
so that the morphism ρIis generically finite. As explained in 2.3 and 2.4, we would like
ΨI(a,Flags(T Ha)|MI)to avoid the exceptional locus of ρIfor abelonging to an open
set of S, for every I({0,...,N},|I|< N.
4.2. Avoiding the exceptional locus. In this subsection, we prove the existence of an
open subset VSsuch that for each |I|< N, and for all aV,ΨI(a,Flags(T Ha)|MI)
avoids the exceptional locus of ρI, provided that the δi’s are taken large enough.
4.2.1. A particular case. Let us prove the existence of such an open set in the setting
where λ= (1k). As we will see afterwards, it is enough to conclude the wanted result
in full generality. In this setting, we denote (some notations were already introduced in
(13)):
Flags(T M ) = Grass(s1, T M ) = Grass(k, T M ).
Ψ
I:S×Grass(k, T M )Grass(k+ 1,kNδ1)×...×Grass(k+ 1,kNδc).
G= Grass(k+ 1,kNδ1)×...×Grass(k+ 1,kNδc).
Y={(∆1,...,c, z)G×PN| ∀1ic, i(z) = 0}.
ρ:YGthe first projection.
Observe that the set Sin the previous setting and in this setting can be chosen to be
the same (Cf. 3.3.3). Let us denote
G
(I):={∆ = (∆1,...,c)G|(ρ
I)1(∆) has positive dimension.}
as well as ΦI=ρIΨI. We prove the following lemma:
Lemma 4.2. If δ1,...,δcare larger than N+k(Nk), there exists a non-empty open
subset VSsuch that for each I({0,...,N}with cardinality |I|< N ,
V×Grass(k, (T M )|MI)Φ
I
1(G
(I)) = .
Proof. We first note that it is enough to prove that for each I({0,...,N}with cardi-
nality Nk,k1, there exists a non-empty open subset V(I)Ssuch that
V(I)×Grass(k, (T M )|MI)Φ
I
1(G
(I)) = ,
since taking their intersection will provide a non-empty open subset satisfying the re-
quired condition. We will prove that
dim (S×Grass(k, (T M )|MI)) Φ
I
1(G
(I))<dim S,
July 23, 2019 24
which will imply the existence of the sought open subset. We fix ηGrass(k, (T M )|MI),
and we denote Φ
η:SGthe map
Φ
η(a) = Φ
I(a, η).
We prove the following inequality
dim Φ
η
1(G
(I))<dim Sdim Grass(k, T M|MI),
which will imply the lemma, since then
dim (S×Grass(k, (T M )|MI)) Φ1(G
(I))
dim Grass(k, T M|MI) + max
ηdim Φ
η
1(G
(I))
<dim S.
In order to study dim Φ